Equation of state and quantal crossover in the transverse ising model

K. Lukierska‐Walasek, W. Salejda

K. Lukierska‐Walasek

Pierre M. Pfeuty

Ian D. Lawrie, Michael Ellis Fisher

The critical properties of d-dimensional Ising models in a transverse magnetc field, Hx=Γ, are considered. A quantal generalization of the Landau-Ginzburg-Wilson free-energy functional, introduced by Hertz and Young, has been treated by a variational approximation of the Hartree type. Behavior near the critcal line, Γ=Γc(T), is governed by the exponents of the d-dimensional spherical model. However, quantal crossover occurs on approach to the T=0 multicritical point at Γ=Γc(0) and this is characterized by the exponents of the (d+1) -dimensional spherical model (which are mean-field-like for d+1≳4). Crossover scaling functions, valid near the multicritical point, are reported for the susceptibility. The quantal crossover exponent φT, is equal to νd+1=1/(d−1), the (d+1) -dimensional correlation length exponent for 2⩽d⩽3, as suggested by ideas of Suzuki, but for 3<d⩽4 we find instead φT= (1/2)(4−d); for d≳4 no crossover occurs.

F.C. SāBarreto, I. P. Fittipaldi

Ian D. Lawrie

The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.